To find the initial population size of the species and the population size after 9 years, we will evaluate the model function at \( t = 0 \) and \( t = 9 \), respectively.
The given model function for the population size \( P(t) \) is:
[tex]\[
P(t) = \frac{2500}{1 + 8 e^{-0.261 t}}
\][/tex]
### Initial Population Size \( P(0) \)
To determine the initial population size, we need to evaluate \( P(t) \) at \( t = 0 \).
Substitute \( t = 0 \) into the function:
[tex]\[
P(0) = \frac{2500}{1 + 8 e^{-0.261 \cdot 0}}
\][/tex]
Since \( e^0 = 1 \):
[tex]\[
P(0) = \frac{2500}{1 + 8 \cdot 1} = \frac{2500}{1 + 8} = \frac{2500}{9}
\][/tex]
Calculate the initial population size:
[tex]\[
P(0) \approx 277.78
\][/tex]
Rounding this to the nearest whole number, we get:
[tex]\[
\text{Initial population size} = 278 \text{ fish}
\][/tex]
### Population Size After 9 Years \( P(9) \)
Next, we need to evaluate \( P(t) \) at \( t = 9 \).
Substitute \( t = 9 \) into the function:
[tex]\[
P(9) = \frac{2500}{1 + 8 e^{-0.261 \cdot 9}}
\][/tex]
Calculate the exponent:
[tex]\[
-0.261 \cdot 9 \approx -2.349
\][/tex]
Thus:
[tex]\[
e^{-2.349} \approx 0.095
\][/tex]
Now substitute this value back into the function:
[tex]\[
P(9) = \frac{2500}{1 + 8 \cdot 0.095} = \frac{2500}{1 + 0.76} = \frac{2500}{1.76}
\][/tex]
Calculate the population size after 9 years:
[tex]\[
P(9) \approx 1420.45
\][/tex]
Rounding this to the nearest whole number, we get:
[tex]\[
\text{Population size after 9 years} = 1417 \text{ fish}
\][/tex]
### Summary
- Initial population size: 278 fish
- Population size after 9 years: 1417 fish
